
where \(B\)  is some generic set in the support of \(Y_{n+1}\)  and \(\mathbb {Y}_{y^n}(v)\) , for \(v \in I_{n+1}\) , is as in (REF ).
[1]} interprets these in a “best of both worlds” sense: on one hand, these are genuine post-data lower and upper probabilities and, therefore, inherit certain coherence properties [2]}; on the other hand, through their connection to the underlying data-generating process through Hill's assumption, they inherit certain calibration properties [3]}. It turns out that these lower and upper probabilities can also be obtained from the above IM construction. Indeed, if we take \(\mathcal {S}= \lbrace \tilde{V}\rbrace \) , for \(\tilde{V} \sim {\sf Unif}(I_{n+1})\) , then the induced distribution of the data-dependent random set \(\mathbb {Y}_{y^n}(\mathcal {S})\)  generates the lower and upper probabilities defined above. Although the aforementioned lower and upper probability approach is a special case of that presented above, there are some key differences between the two, resulting from our proposed random set \(\mathcal {S}\)  being nested and, therefore, structurally different from the non-nested singleton random sets that generate the \((\underline{\mathsf {P}}, \overline{\mathsf {P}})\)  output. In particular, there is no useful sense in which once can assess the “plausibility” that the next observation, \(Y_{n+1}\) , will be equal to some specified \(y\) : the lower probability equals 0 and the upper probability equals \((n+1)^{-1}\)  for every \(y\) . Compare this to the plausibility contour displayed in Figure REF (b), where there is one \(y\)  which is the “most plausible” value of \(Y_{n+1}\)  based on the observed data \(y^n\) . Having this plausibility contour makes reading off prediction intervals with desired coverage level straightforward. Of course, this same calibration property is embedded in the lower and upper probabilities above, but it is more difficult to extract when represented in that form. To our knowledge, given a lower probability like that above, the most direct way to extract a nominal \(100(1-\alpha )\) % prediction interval is via the intersection
\( \bigcap \bigl \lbrace B \subseteq \mathbb {R}: \underline{\mathsf {P}}(Y_{n+1} \in B \mid y^n) \ge 1-\alpha \bigr \rbrace . \) 
